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The concept of statistical convergence was presented by Steinhaus in 1951. This concept was extended to the double sequences by Mursaleen and Edely in 2003. Karakus has recently introduced the concept of statistical convergence of ordinary (single) sequence on probabilistic normed spaces. In this paper, we define statistical analogues of convergence and Cauchy for double sequences on probabilistic normed spaces. Then we display an example such that our method of convergence is stronger than usual convergence on probabilistic normed spaces. Also we give a useful characterization for statistically convergent double sequences.

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Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715-729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.

We deal with a new type of statistical convergence for double sequences, calledΨ-A-statistical convergence, and we prove a Korovkin-type approximation theorem with respect to this type of convergence in modular spaces. Finally, we give some application to moment-type operators in Orlicz spaces.

Abstract. In this paper we extend the concepts of statistical limit superior and inferior (as introduced by FVidy and Orhan) to A-statistical limit superior and inferior and give some .¿-statistical analogue of properties of statistical limit superior and inferior for a sequence of real numbers. Also we extend the concept of statistical core to A-statistical core and get necessary and sufficient conditions on a matrix T so that the Knopp core of Tx is contained in the Astatistical core of a bounded complex number sequence x. IntroductionIf A" is a set of positive integers, K n will denote the set {k € K : k < n} and \K n \ will denote the cardinality of K n . The natural density of K [16] is given by 6{K) := lim" if it exists. The number sequence x = (x*) is statistically convergent to L provided that for every e > 0 the set K := K(e) := {k € N : |x* -L\ > «} has natural density zero [6], [8]. In this case, we write st -limx = L. Hence x is statistically convergent to L iff Statistical convergence can be generalized by using a regular nonnegative summability matrix A in place of Ci.Following Freedman and Sember [7], we say that a set K Ç N has. A-density if 6A(K) := lim"(Axjr)n = limn 52K€K ^fc exists where A is a nonnegative regular matrix.The number sequence x = (x*) is A-statistically convergent to L provided that for every e > 0 the set K(e) has A-density zero [3]. In this case we write SÎA -limx = L. This idea was studied by Freedman and Sember [7]

In the present paper, using the concept of statistical relative convergence, we study the problem of approximation to a function by means of double sequences of positive linear operators defined on a modular space. Also, a non-trivial application is presented.Mathematics Subject Classification. 40B05, 41A36, 46E30.